Average Error: 3.7 → 2.3
Time: 11.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.1856454375998788 \cdot 10^{206}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(2, \frac{1}{{\alpha}^{2}}, 1 - 1 \cdot \frac{1}{\alpha}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.1856454375998788 \cdot 10^{206}:\\
\;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(2, \frac{1}{{\alpha}^{2}}, 1 - 1 \cdot \frac{1}{\alpha}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\end{array}
double f(double alpha, double beta) {
        double r109792 = alpha;
        double r109793 = beta;
        double r109794 = r109792 + r109793;
        double r109795 = r109793 * r109792;
        double r109796 = r109794 + r109795;
        double r109797 = 1.0;
        double r109798 = r109796 + r109797;
        double r109799 = 2.0;
        double r109800 = r109799 * r109797;
        double r109801 = r109794 + r109800;
        double r109802 = r109798 / r109801;
        double r109803 = r109802 / r109801;
        double r109804 = r109801 + r109797;
        double r109805 = r109803 / r109804;
        return r109805;
}
↓
double f(double alpha, double beta) {
        double r109806 = alpha;
        double r109807 = 1.1856454375998788e+206;
        bool r109808 = r109806 <= r109807;
        double r109809 = beta;
        double r109810 = r109806 + r109809;
        double r109811 = r109809 * r109806;
        double r109812 = r109810 + r109811;
        double r109813 = 1.0;
        double r109814 = r109812 + r109813;
        double r109815 = 2.0;
        double r109816 = r109815 * r109813;
        double r109817 = r109810 + r109816;
        double r109818 = r109814 / r109817;
        double r109819 = r109818 / r109817;
        double r109820 = log1p(r109819);
        double r109821 = expm1(r109820);
        double r109822 = r109817 + r109813;
        double r109823 = r109821 / r109822;
        double r109824 = 1.0;
        double r109825 = 2.0;
        double r109826 = pow(r109806, r109825);
        double r109827 = r109824 / r109826;
        double r109828 = r109824 / r109806;
        double r109829 = r109813 * r109828;
        double r109830 = r109824 - r109829;
        double r109831 = fma(r109815, r109827, r109830);
        double r109832 = r109831 / r109817;
        double r109833 = log1p(r109832);
        double r109834 = expm1(r109833);
        double r109835 = r109834 / r109822;
        double r109836 = r109808 ? r109823 : r109835;
        return r109836;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.1856454375998788e+206

    1. Initial program 1.9

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
    2. Using strategy rm

    3. Applied expm1-log1p-u1.9

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    if 1.1856454375998788e+206 < alpha

    1. Initial program 17.8

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
    2. Using strategy rm

    3. Applied expm1-log1p-u17.8

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    4. Taylor expanded around inf 5.4

      \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\left(2 \cdot \frac{1}{{\alpha}^{2}} + 1\right) - 1 \cdot \frac{1}{\alpha}}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]

    5. Simplified5.4

      \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(2, \frac{1}{{\alpha}^{2}}, 1 - 1 \cdot \frac{1}{\alpha}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.1856454375998788 \cdot 10^{206}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(2, \frac{1}{{\alpha}^{2}}, 1 - 1 \cdot \frac{1}{\alpha}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))